# American Institute of Mathematical Sciences

March  2016, 36(3): 1693-1707. doi: 10.3934/dcds.2016.36.1693

## Structurally stable homoclinic classes

 1 School of Mathematics and Systems Science, Beihang University, Beijing 100191

Received  October 2014 Revised  June 2015 Published  August 2015

In this paper we study structurally stable homoclinic classes. In a natural way, the structural stability for an individual homoclinic class is defined through the continuation of periodic points. Since the classes is not innately locally maximal, it is hard to answer whether structurally stable homoclinic classes are hyperbolic. In this article, we make some progress on this question. We prove that if a homoclinic class is structurally stable, then it admits a dominated splitting. Moreover we prove that codimension one structurally stable classes are hyperbolic. Also, if the diffeomorphism is far away from homoclinic tangencies, then structurally stable homoclinic classes are hyperbolic.
Citation: Xiao Wen. Structurally stable homoclinic classes. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1693-1707. doi: 10.3934/dcds.2016.36.1693
##### References:
 [1] F. Abdenur, Ch. Bonatti, S. Crovisier, L. J. Diáz and L. Wen, Periodic points and homoclinic classes,, Ergodic Theory and Dynamical Systems, 27 (2007), 1. doi: 10.1017/S0143385706000538. Google Scholar [2] C. Bonatti and S. Crovisier, Recurrence and genericity,, Invent. Math., 158 (2004), 33. doi: 10.1007/s00222-004-0368-1. Google Scholar [3] C. Bonatti L. Díaz, and E. Pujals, A $C^1$-generic dichotomy for diffeomorphisms: Weak forms of hyperbolicity or infinitely many sinks or sources,, Ann. of Math., 158 (2003), 355. doi: 10.4007/annals.2003.158.355. Google Scholar [4] C. Bonatti, S. Gan and D. Yang, On the hyperbolicity of homoclinic classes,, Discrete Contin. Dyn. Syst., 25 (2009), 1143. doi: 10.3934/dcds.2009.25.1143. Google Scholar [5] S. Crovisier, Partial hyperbolicity far from homoclinic bifurcations,, Advances in Mathematics, 226 (2011), 673. doi: 10.1016/j.aim.2010.07.013. Google Scholar [6] S. Crovisier, Periodic orbits and chain-transitive sets of $C^1$-diffeomorphisms,, Publ. Math. I.H.E.S, 104 (2006), 87. doi: 10.1007/s10240-006-0002-4. Google Scholar [7] S. Crovisier, M. Sambarino and D. Yang, Partial hyperbolicity and homoclinic tangencies,, Journal of the European Mathematical Society, 17 (2015), 1. doi: 10.4171/JEMS/497. Google Scholar [8] J. Franks, Necessary conditions for stability of diffeomorphisms,, Transactions of the A.M.S 158 (1971), 158 (1971), 301. doi: 10.1090/S0002-9947-1971-0283812-3. Google Scholar [9] M. Hirsch, C. Pugh and M. Shub, Invariant Manifolds,, Lect. Notes in Math., (1977). Google Scholar [10] S. Gan and L. Wen, Heteroclinic cycles and homoclinic closures for generic diffeomorphisms,, J. Dynam. Differential Equations, 15 (2003), 451. doi: 10.1023/B:JODY.0000009743.10365.9d. Google Scholar [11] N. Gourmelon, An Isotopic Perturbation Lemma Along Periodic Orbits manifolds,, preprint, (). Google Scholar [12] S. Liao, A basic property of a certain class of differential systems,, Acta Math. Sinica, 22 (1979), 316. Google Scholar [13] S. Liao, Obstruction sets II,, Acta Sci. Natur. Univ. Pekinensis, 2 (1981), 1. Google Scholar [14] R. Mañé, An ergodic closing lemma,, Annals of Math., 116 (1982), 503. doi: 10.2307/2007021. Google Scholar [15] R. Mañé, A proof of the $C^1$ stability conjecture,, Publ. Math. I.H.E.S, 66 (1988), 161. Google Scholar [16] J. Palis and W. de Melo, Geometric Theory of Dynamical Systems. An Introduction,, Transl. from the Portuguese by A. K. Manning. (English), (1982). Google Scholar [17] J. Palis and S. Smale, Structural stability theorems,, in Global Analysis (Proc. Sympos. Pure Math., (1968), 223. Google Scholar [18] E. Pujals and M. Sambarino, Homoclinic tangencies and hyperbolicity for surface diffeomorphisms,, Annals of Math., 151 (2000), 961. doi: 10.2307/121127. Google Scholar [19] M. Sambarino and J. Vieitez, On $C^1$-persistently expansive homoclinic classes,, Discrete Contin. Dyn. Syst., 14 (2006), 465. Google Scholar [20] X. Wen and L. Wen, Codimension One Structurally Stable Chain Classes,, to appear in Transactions of the A.M.S., (). doi: 10.1090/tran/6440. Google Scholar

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##### References:
 [1] F. Abdenur, Ch. Bonatti, S. Crovisier, L. J. Diáz and L. Wen, Periodic points and homoclinic classes,, Ergodic Theory and Dynamical Systems, 27 (2007), 1. doi: 10.1017/S0143385706000538. Google Scholar [2] C. Bonatti and S. Crovisier, Recurrence and genericity,, Invent. Math., 158 (2004), 33. doi: 10.1007/s00222-004-0368-1. Google Scholar [3] C. Bonatti L. Díaz, and E. Pujals, A $C^1$-generic dichotomy for diffeomorphisms: Weak forms of hyperbolicity or infinitely many sinks or sources,, Ann. of Math., 158 (2003), 355. doi: 10.4007/annals.2003.158.355. Google Scholar [4] C. Bonatti, S. Gan and D. Yang, On the hyperbolicity of homoclinic classes,, Discrete Contin. Dyn. Syst., 25 (2009), 1143. doi: 10.3934/dcds.2009.25.1143. Google Scholar [5] S. Crovisier, Partial hyperbolicity far from homoclinic bifurcations,, Advances in Mathematics, 226 (2011), 673. doi: 10.1016/j.aim.2010.07.013. Google Scholar [6] S. Crovisier, Periodic orbits and chain-transitive sets of $C^1$-diffeomorphisms,, Publ. Math. I.H.E.S, 104 (2006), 87. doi: 10.1007/s10240-006-0002-4. Google Scholar [7] S. Crovisier, M. Sambarino and D. Yang, Partial hyperbolicity and homoclinic tangencies,, Journal of the European Mathematical Society, 17 (2015), 1. doi: 10.4171/JEMS/497. Google Scholar [8] J. Franks, Necessary conditions for stability of diffeomorphisms,, Transactions of the A.M.S 158 (1971), 158 (1971), 301. doi: 10.1090/S0002-9947-1971-0283812-3. Google Scholar [9] M. Hirsch, C. Pugh and M. Shub, Invariant Manifolds,, Lect. Notes in Math., (1977). Google Scholar [10] S. Gan and L. Wen, Heteroclinic cycles and homoclinic closures for generic diffeomorphisms,, J. Dynam. Differential Equations, 15 (2003), 451. doi: 10.1023/B:JODY.0000009743.10365.9d. Google Scholar [11] N. Gourmelon, An Isotopic Perturbation Lemma Along Periodic Orbits manifolds,, preprint, (). Google Scholar [12] S. Liao, A basic property of a certain class of differential systems,, Acta Math. Sinica, 22 (1979), 316. Google Scholar [13] S. Liao, Obstruction sets II,, Acta Sci. Natur. Univ. Pekinensis, 2 (1981), 1. Google Scholar [14] R. Mañé, An ergodic closing lemma,, Annals of Math., 116 (1982), 503. doi: 10.2307/2007021. Google Scholar [15] R. Mañé, A proof of the $C^1$ stability conjecture,, Publ. Math. I.H.E.S, 66 (1988), 161. Google Scholar [16] J. Palis and W. de Melo, Geometric Theory of Dynamical Systems. An Introduction,, Transl. from the Portuguese by A. K. Manning. (English), (1982). Google Scholar [17] J. Palis and S. Smale, Structural stability theorems,, in Global Analysis (Proc. Sympos. Pure Math., (1968), 223. Google Scholar [18] E. Pujals and M. Sambarino, Homoclinic tangencies and hyperbolicity for surface diffeomorphisms,, Annals of Math., 151 (2000), 961. doi: 10.2307/121127. Google Scholar [19] M. Sambarino and J. Vieitez, On $C^1$-persistently expansive homoclinic classes,, Discrete Contin. Dyn. Syst., 14 (2006), 465. Google Scholar [20] X. Wen and L. Wen, Codimension One Structurally Stable Chain Classes,, to appear in Transactions of the A.M.S., (). doi: 10.1090/tran/6440. Google Scholar
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